Starting from a suitable sequence of auto-Poisson lattice schemes, it is shown that (almost) any purely inhibitory pairwise-interaction point process can be obtained in the limit. Further pairwise-interaction processes are obtained as limits of sequences of auto-logistic lattice schemes.

We obtain an extension of a theorem of Bingham and Doney connecting the random variables Z 1 and W in the supercritical Galton-Watson process. The regular variation of the distribution of Z1 is equivalent to the regular variation of the tail of the distribution of W for integer values of α > 1.

If X = (X 1, · ··, Xn ) has uniform distribution on the sphere or ball in ℝ with radius a, then the joint distribution of , ···, k, converges in total variation to the standard normal distribution on ℝ. Similar results hold for the inner products of independent n-vectors. Applications to geometric probability are given.

The properties of a measure of randomness on the class of discrete probability distributions are discussed. A result concerning finiteness is obtained for a family of measures which includes the entropy function.

The limit distribution of the k th maximum from a random sample of size n when n → ∞ is identified as the distribution of the k th lower record value from one of three extreme value distributions. This fact is used in giving a different canonical representation and new proofs of the results of Hall (1978) for this limiting random variable. A characterization of the exponential distribution based on upper record values is given.

The present value is studied when I(t) is a stationary random income stream. The stationary distribution of V(t) for a family of simple streams modeled by stationary finite Markov chains is given explicitly. The process V(t) is shown to be observable in a special sense when I(t) is time-reversible.

Ross (1978) conjectured that the average delay in a single-server queue is larger when the arrival process is a non-stationary Poisson process than when it is a stationary Poisson process with the same rate. We present an example where equality obtains. When the number of waiting-positions is finite, Ross conjectured that the proportion of lost customers is greater in the nonstationary case. We present a counterexample to this conjecture.